On a class of holomorphic functions representable by Carleman formulas in the disk from their values on the arc of the circle

✍ Scribed by Lev Aizenberg; Alekos Vidras

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 218 KB
• Volume: 280
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200410460

✦ Synopsis

Abstract

Let D be a unit disk and__M__ be an open arc of the unit circle whose Lebesgue measure satisfies 0 < l (M) < 2__π__. Our first result characterizes the restriction of the holomorphic functions f ∈ ℋ︁(D), which are in the Hardy class ℋ︁^1^ near the arc__M__ and are denoted by 𝒩 ℋ︁^1^~M~ (𝒟), to the open arc__M__. This result is a direct consequence of the complete description of the space of holomorphic functions in the unit disk which are represented by the Carleman formulas on the open arc M.

As an application of the above characterization, we present an extension theorem for a function f ∈ L ^1^(M) from any symmetric sub‐arc L ⊂ M of the unit circle, such that $ \bar L $ ⊂ M, to a function f ∈ 𝒩 ℋ︁^1^~L~ (𝒟). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)